Vanishing theorems on Hermitian manifolds

1999/01/31 by Bogdan Alexandrov, Stefan Ivanov · 2 citations
Mathematics · #math.DG #msc:53C55 #msc:53C15

paper · pdf

published as Differ. Geom. Appl. 14, No.3, 251-265 (2001) · 15 pages, Latex format, no figures; Added section

arxiv created 2003/01/09 · arxiv updated 2009/11/30

Abstract

We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with dd^c-harmonic Kähler form and positive (1,1)-part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Hermitian metrics, such that the Ricci tensor of the canonical Weyl structure is positive. As a corollary we obtain that any such surface must be rational with c₁² >0. As an application, the pth Dolbeault cohomology groups of a left-invariant complex structure compatible with a bi-invariant metric on a compact even dimensional Lie group are computed.

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• Positive Hermitian curvature flow on 2-step nilpotent Lie groups
• Almost Calabi-Yau with torsion 6-manifolds and the instanton condition

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